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Finite element analysis of shells of revolution
Book Reviews
357
the reviewer does recommend this book to those interested in acquiring a good understanding of FE in fluid mechanics. H. SAUNDERS 1 Arcadian Drive Scotia, N Y 12302, U.S.A.
"Finite Element Analysis of Shells of Revolution", by P.L. Gould, Pitman Pub. Co., distributed by Longman's, Inc., 95 Church Street, White Plains, New York, NY, U.S.A., 1985, 210 pp., ISBN 0-273-08654-5 This little volume packs a 'tremendous wallop'. The monograph contains a great deal of information enclosed within its covers. The book stresses the finite element (FE) analysis on shells of revolution, with emphasis on the geometrical and structural forms cast in FE. Each component in this procedure treats the geometry, loading and mass in exact theoretical fashion. The author produces a concise methodology for the analysis of shells of revolution in both statics and dynamics plus thermal axisymmetrical loading. The shell of revolution is supported on elastic boundaries. As stated by the author: " T h e emphasis is in linear problems, both statics and dynamics, since most engineering applications fall within this regime; however, many of the concepts are applicable to nonlinear problems. Considerable emphasis is placed on the techniques of modeling, with detailed discussion of discretization, representation of loading and interpretation of results. This is carried out mainly through commentaries within the numerical case histories." A number of the derived equations are stated in FE and applied to his computer program SHORE III. This program is constantly being expanded and upgraded. The text consists of seven chapters. Chapter 1 introduces the subject and provides a short historical review of FE. Chapter 2 expresses the fundamentals. Beginning with the surface geometry, related by using the Gauss-Codozzi equation, this progresses ahead into derivation of ring element using Lagrangian polynomials. We proceed with the kinematic and statical variables of shells of revolution and representation of shells of revolution dependent variables into Fourier series. This follows with transverse shearing strains in linear form with a short discourse in nonlinear guise and the constitutive equations relating the possibility of accommodating shells with complicated material properties. This continues with boundary conditions applied to shells. Our next topic is variational principles which provides a convenient way of introducing approximations associated with the FE method. This covers Reissner's Principle, Hamilton's Principle and mention of the Rayleigh-Ritz method. Chapter 3 focusses on static analyses. The initial topic is the linear displacement formulation. We employ the element equilibrium equations, set in matrix form. In order to reduce the number of equations, FE usually employs static condensation. The author applies the assembly of the global equilibrium equation by means of the direct stiffness method with the proper boundary conditions. The solution depends upon strains, stresses, and rigid body displacements. The basic element is then extended to stiffened shells which encompass circumferential and meridional (stringers) stiffeners. With this under our belt, the book reflects compound and branched shells plus open-type elements. There is a prime need to transfer inclined members and slope discontinuity from local to global coordinates. When members of an open-type element are fully continuous with the adjacent shell elements, the normal rotational degrees of freedom are neglected. This reduces the stiffness matrix and no condensation is necessary.
358
Book Reviews
Next, the book shows how consistent load vectors are developed and introduced into the specialized element stiffness matrix. This leads to convergence and discretization criteria where one improves the mesh arrangemen in a systematic fashion. The book continues with specialized case histories which considers (a) cylindrical shells under edge loading, (b) cylindrical shells under hydrostatic loading, (c) parabola shell under symmetric loading, (d) hyperboloid shell under static wind loading, (e) cylindrical shell with thermal loading, and (f) cylindrical shell with torispherical head. The SHORE program solution is compared with the BOSOR computer program solution or test results were applicable. The next chapter plunges ahead into dynamic analysis. The uncondensed equations of motion are initially given and then assembled into global form. This is extended to the consistent mass matrix for an open-type element followed by the modal solution of free vibration. The next topic is response system analysis which incorporates the complex response method and proceeds to a vary concise discussion of direct integration method of solution. Case histories cover (a) free vibrations of cylindrical shell, (b) free vibration of a hemispherical shell, and (c) dynamic analysis of a column supported colling lower shell. Application of the latter is represented by the response spectra when consideration is given to the rocking of the foundation on soil and pile. The chapter concludes with a cylindrical shell under blast load, hyperboloidal shell under dynamic wind load and free vibrations of a fluid-filled cylindrical shell. An excellent chapter that should be carefully read! Chapter 5 explores geometric nonlinearities and instability. The opening section considers strain displacement relations, modified equilibrium condition, and evaluation of the geometric stiffness matrix for the shell of revolution, and ignores harmonic coupling. We step forward into the displacement interpolation function with special emphasis on first-order interpolation. Bifurcation buckling is next on the agenda with direct implication to a hyperbolic cooling tower shell. The concluding section dwells upon the effect of imperfection and employment of incremental nonlinear analysis in determining the critical or buckling load. Chapter 6 focusses upon the analysis of locally nonaxisymmetric shells which contain certain local irregularities. The general shell element is patterned after the isoparametric element and follows with the transitional element. We have a given element bordering on one side, a rotational element on the opposite side which then joins transitional elements on the remaining two sides. The author describes the global system pattern and elucidates the proper solution procedure. Case studies include (a) imperfect hyperboloid shell under self weight and wing loading, and (b) cylindrical shell with a circular cutout. The final chapter competes and dissects the BOSOR 4 computer program (finite difference) and the SHORE III FE program. The former performs stress, stability, and vibration analysis of segmented branched ring stiffened elastic shells of revolution with various end conditions. SHORE III is designed for static and dynamic analysis of shells of revolution. As in previous chapters, SHORE III addresses itself to a case study of a hyperboloidal shell on column supports. The shell is analyzed for self weight, static wind load, and earthquake loading stress couples. Stress resultants are computed at critical locations around the circumference and throughout the height. In summary, this is an excellent book. The author furnishes good definitions and symbols for matrices. He also includes the Hennitian interpolition function and good use of the Lagrangian multiplier. The reviewer would have preferred seeing a table of abbreviations used in the book plus a more extended section on isoparametric elements using the 20-node element. Portions of the SHORE III computer program should be included as well as examples showing an expended version of Newmark's method, Wilson's method, and Houbolt's method applied to tune integration method. A prime example of stress concentration factors which could be determined by FE analysis and thus add flavor to the book would be a three-dimensional analysis of two intersecting circular pipes. Nevertheless, this omission should not detract from
Book Reviews
359
the book. The reviewer heartily recommends this book to those interested in the FE analysis of shells of revolution. H. SAUNDERS 1 Arcadian Drive Scotia, N Y 12302, U.S.A.
"Seismic Effects in PVP Components", edited by V.N. Shaw and D.C. Ma, PVP, Vol. 188, ASME, New York, NY, U.S.A., 1984, 109 pp. Seismic analysis is an ongoing program. Recent mishaps at nuclear power plants have focussed great attention upon their safety. During the early years of this industry, design effort was slanted more towards 'seat of the pants' analysis. Within the last 2½ decades, analysis has become a much more 'in-depth' process-nothing is left to the 'seat of the pants'. Finite element (FE) has been and is playing an important role in nuclear power plant design. Design analysis requires that FE and testing be 'blood brothers'. The editors of this volume state that: " T h e papers in this symposium (June 1984) may be divided into two groups. The first group of four papers presents the computational methods, mathematical modeling and vibration test results to upgrade the correct seismic analysis methods . . . . The second group of four papers contains the seismic analysis and testing of liquid-filled systems, such as ground supported or liquid storage tanks that are buried and reactor primary tanks." The initial paper reports on modal combination methods in the response spectrum analysis of piping systems. The two important effects are (a) residual rigid response or higher modal effect (HME), and (b) correlation between modal and rigid responses. The author's previous analysis takes these effects into account. He further proves that thus method provides answers which almost coincide with direct integration analysis results. They are superior to previous model combination rules. The author employs FE method in applying it to a piping system. Paper # 2 deals with equipment modeling in piping dynamic analysis. The response of piping systems to an earthquake or aircraft crash is performed. Usually, it is investigated by floor response spectra. The two proposed systems are (a) generating nozzle response spectra, and (b) introduction of simplified equipment models in the piping analysis. The author shows that including the equipment models in the piping system results in a valuable contribution to the system behavior. It is more reasonable than only relying on nozzle response spectra. The more sophisticated in this paper increase hardware, software, management, and QA costs. Paper # 3 covers structural damping results from vibration tests of straight piping section. A series of vibration tests were conducted in order to ascertain any alterations in structural damping due to various parameter effects. Spring, rod and constant force hangers as well as sway braces and snubbers were some major components of these tests. Damping was higher in those configurations having loose fittings. Spring hangers kept tension on the loose connections and virtually eliminated any chatter. Paper # 4 considers the implication of the response resulting from impact of components occurring in a seismic event of the nuclear power plant. In order to identify these critical cases, a parameter variation of the mass, input and critical frequency range of the components is employed. Based upon the analysis, the frequent characteristics play an important part in defining the critical range of the components. The authors employ the FE method with a gap element in the analysis. Paper # 5 addresses the analytical and experimental investigations in the fluid structure interaction in pool type LMFBR's plus an examination of the validity of analytical models in fluid structure coupled problems. A simplified added mass and FE analysis were conducted. Based on tests, the
357
the reviewer does recommend this book to those interested in acquiring a good understanding of FE in fluid mechanics. H. SAUNDERS 1 Arcadian Drive Scotia, N Y 12302, U.S.A.
"Finite Element Analysis of Shells of Revolution", by P.L. Gould, Pitman Pub. Co., distributed by Longman's, Inc., 95 Church Street, White Plains, New York, NY, U.S.A., 1985, 210 pp., ISBN 0-273-08654-5 This little volume packs a 'tremendous wallop'. The monograph contains a great deal of information enclosed within its covers. The book stresses the finite element (FE) analysis on shells of revolution, with emphasis on the geometrical and structural forms cast in FE. Each component in this procedure treats the geometry, loading and mass in exact theoretical fashion. The author produces a concise methodology for the analysis of shells of revolution in both statics and dynamics plus thermal axisymmetrical loading. The shell of revolution is supported on elastic boundaries. As stated by the author: " T h e emphasis is in linear problems, both statics and dynamics, since most engineering applications fall within this regime; however, many of the concepts are applicable to nonlinear problems. Considerable emphasis is placed on the techniques of modeling, with detailed discussion of discretization, representation of loading and interpretation of results. This is carried out mainly through commentaries within the numerical case histories." A number of the derived equations are stated in FE and applied to his computer program SHORE III. This program is constantly being expanded and upgraded. The text consists of seven chapters. Chapter 1 introduces the subject and provides a short historical review of FE. Chapter 2 expresses the fundamentals. Beginning with the surface geometry, related by using the Gauss-Codozzi equation, this progresses ahead into derivation of ring element using Lagrangian polynomials. We proceed with the kinematic and statical variables of shells of revolution and representation of shells of revolution dependent variables into Fourier series. This follows with transverse shearing strains in linear form with a short discourse in nonlinear guise and the constitutive equations relating the possibility of accommodating shells with complicated material properties. This continues with boundary conditions applied to shells. Our next topic is variational principles which provides a convenient way of introducing approximations associated with the FE method. This covers Reissner's Principle, Hamilton's Principle and mention of the Rayleigh-Ritz method. Chapter 3 focusses on static analyses. The initial topic is the linear displacement formulation. We employ the element equilibrium equations, set in matrix form. In order to reduce the number of equations, FE usually employs static condensation. The author applies the assembly of the global equilibrium equation by means of the direct stiffness method with the proper boundary conditions. The solution depends upon strains, stresses, and rigid body displacements. The basic element is then extended to stiffened shells which encompass circumferential and meridional (stringers) stiffeners. With this under our belt, the book reflects compound and branched shells plus open-type elements. There is a prime need to transfer inclined members and slope discontinuity from local to global coordinates. When members of an open-type element are fully continuous with the adjacent shell elements, the normal rotational degrees of freedom are neglected. This reduces the stiffness matrix and no condensation is necessary.
358
Book Reviews
Next, the book shows how consistent load vectors are developed and introduced into the specialized element stiffness matrix. This leads to convergence and discretization criteria where one improves the mesh arrangemen in a systematic fashion. The book continues with specialized case histories which considers (a) cylindrical shells under edge loading, (b) cylindrical shells under hydrostatic loading, (c) parabola shell under symmetric loading, (d) hyperboloid shell under static wind loading, (e) cylindrical shell with thermal loading, and (f) cylindrical shell with torispherical head. The SHORE program solution is compared with the BOSOR computer program solution or test results were applicable. The next chapter plunges ahead into dynamic analysis. The uncondensed equations of motion are initially given and then assembled into global form. This is extended to the consistent mass matrix for an open-type element followed by the modal solution of free vibration. The next topic is response system analysis which incorporates the complex response method and proceeds to a vary concise discussion of direct integration method of solution. Case histories cover (a) free vibrations of cylindrical shell, (b) free vibration of a hemispherical shell, and (c) dynamic analysis of a column supported colling lower shell. Application of the latter is represented by the response spectra when consideration is given to the rocking of the foundation on soil and pile. The chapter concludes with a cylindrical shell under blast load, hyperboloidal shell under dynamic wind load and free vibrations of a fluid-filled cylindrical shell. An excellent chapter that should be carefully read! Chapter 5 explores geometric nonlinearities and instability. The opening section considers strain displacement relations, modified equilibrium condition, and evaluation of the geometric stiffness matrix for the shell of revolution, and ignores harmonic coupling. We step forward into the displacement interpolation function with special emphasis on first-order interpolation. Bifurcation buckling is next on the agenda with direct implication to a hyperbolic cooling tower shell. The concluding section dwells upon the effect of imperfection and employment of incremental nonlinear analysis in determining the critical or buckling load. Chapter 6 focusses upon the analysis of locally nonaxisymmetric shells which contain certain local irregularities. The general shell element is patterned after the isoparametric element and follows with the transitional element. We have a given element bordering on one side, a rotational element on the opposite side which then joins transitional elements on the remaining two sides. The author describes the global system pattern and elucidates the proper solution procedure. Case studies include (a) imperfect hyperboloid shell under self weight and wing loading, and (b) cylindrical shell with a circular cutout. The final chapter competes and dissects the BOSOR 4 computer program (finite difference) and the SHORE III FE program. The former performs stress, stability, and vibration analysis of segmented branched ring stiffened elastic shells of revolution with various end conditions. SHORE III is designed for static and dynamic analysis of shells of revolution. As in previous chapters, SHORE III addresses itself to a case study of a hyperboloidal shell on column supports. The shell is analyzed for self weight, static wind load, and earthquake loading stress couples. Stress resultants are computed at critical locations around the circumference and throughout the height. In summary, this is an excellent book. The author furnishes good definitions and symbols for matrices. He also includes the Hennitian interpolition function and good use of the Lagrangian multiplier. The reviewer would have preferred seeing a table of abbreviations used in the book plus a more extended section on isoparametric elements using the 20-node element. Portions of the SHORE III computer program should be included as well as examples showing an expended version of Newmark's method, Wilson's method, and Houbolt's method applied to tune integration method. A prime example of stress concentration factors which could be determined by FE analysis and thus add flavor to the book would be a three-dimensional analysis of two intersecting circular pipes. Nevertheless, this omission should not detract from
Book Reviews
359
the book. The reviewer heartily recommends this book to those interested in the FE analysis of shells of revolution. H. SAUNDERS 1 Arcadian Drive Scotia, N Y 12302, U.S.A.
"Seismic Effects in PVP Components", edited by V.N. Shaw and D.C. Ma, PVP, Vol. 188, ASME, New York, NY, U.S.A., 1984, 109 pp. Seismic analysis is an ongoing program. Recent mishaps at nuclear power plants have focussed great attention upon their safety. During the early years of this industry, design effort was slanted more towards 'seat of the pants' analysis. Within the last 2½ decades, analysis has become a much more 'in-depth' process-nothing is left to the 'seat of the pants'. Finite element (FE) has been and is playing an important role in nuclear power plant design. Design analysis requires that FE and testing be 'blood brothers'. The editors of this volume state that: " T h e papers in this symposium (June 1984) may be divided into two groups. The first group of four papers presents the computational methods, mathematical modeling and vibration test results to upgrade the correct seismic analysis methods . . . . The second group of four papers contains the seismic analysis and testing of liquid-filled systems, such as ground supported or liquid storage tanks that are buried and reactor primary tanks." The initial paper reports on modal combination methods in the response spectrum analysis of piping systems. The two important effects are (a) residual rigid response or higher modal effect (HME), and (b) correlation between modal and rigid responses. The author's previous analysis takes these effects into account. He further proves that thus method provides answers which almost coincide with direct integration analysis results. They are superior to previous model combination rules. The author employs FE method in applying it to a piping system. Paper # 2 deals with equipment modeling in piping dynamic analysis. The response of piping systems to an earthquake or aircraft crash is performed. Usually, it is investigated by floor response spectra. The two proposed systems are (a) generating nozzle response spectra, and (b) introduction of simplified equipment models in the piping analysis. The author shows that including the equipment models in the piping system results in a valuable contribution to the system behavior. It is more reasonable than only relying on nozzle response spectra. The more sophisticated in this paper increase hardware, software, management, and QA costs. Paper # 3 covers structural damping results from vibration tests of straight piping section. A series of vibration tests were conducted in order to ascertain any alterations in structural damping due to various parameter effects. Spring, rod and constant force hangers as well as sway braces and snubbers were some major components of these tests. Damping was higher in those configurations having loose fittings. Spring hangers kept tension on the loose connections and virtually eliminated any chatter. Paper # 4 considers the implication of the response resulting from impact of components occurring in a seismic event of the nuclear power plant. In order to identify these critical cases, a parameter variation of the mass, input and critical frequency range of the components is employed. Based upon the analysis, the frequent characteristics play an important part in defining the critical range of the components. The authors employ the FE method with a gap element in the analysis. Paper # 5 addresses the analytical and experimental investigations in the fluid structure interaction in pool type LMFBR's plus an examination of the validity of analytical models in fluid structure coupled problems. A simplified added mass and FE analysis were conducted. Based on tests, the